You can try searching (google) for the output of lspci or lsusb: like "1234:5678 firmware". Alternatively, search for your chip in the Linux Wireless data base.
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In mint have you tried the latest kernel through update manager ?
Real question: if your WiFi doesn't work and you don't have a wired connection available, what do you do to update the kernel? Can you just download it on another machine and then update with a USB drive?